Matematické Fórum

Skrytý text:

prove that .From AH inequality
we get . For we get (with using the sum of arithmetic series).
prove that . From Cauchy-Schwarz inequality for all vectors x and y of an inner product space, we get
. With using
we get  ( the proof - http://en.wikipedia.org/wiki/Basel_problem ). So  . Surely then  .
(Assume that , then and  it is contradition).
Edit:IN is MISTAKE, then proof is bad.

Skrytý text:

: If then .
Proof: Let , . Then , because and
(by using partial fractions).
So,  .
: If  , then for .
Proof: Let , , (expand by ).
Then , because .So .
By summation from to we get .
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prove that .
From Cauchy-Schwarz inequality for all vectors x and y of an inner product space, we get
.
By using , we get  .
So,.

prove that .
From Cauchy-Schwarz inequality we get
.
By using , we get .
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I hope here aren't so important mistakes like in my first contribution.:)

Skrytý text:
Nice prove Rumburak